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A182299 Number of achiral simplicial 4-clusters with n cells. 3
1, 1, 1, 3, 6, 20, 51, 184, 550, 2009, 6487, 23875, 81724, 302954, 1078409, 4034373, 14771551, 55789188, 208526682, 794933818, 3017839193, 11604938152, 44590911769, 172833268057, 670520982414, 2617397888002, 10234831661388, 40204487779050, 158254659096516, 625142808049902 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
This sequence would be H_{4,n} subtracted from twice h_{4,n} in Table 8 of the Hering article if those numbers were correct, but some are not. In addition, the formula in the penultimate line of Table 5 of the article should not have an exponent for the second E_4.
The limit supremum as n approaches infinity of a(n+1)/a(n) is 16(2-sqrt(3)) or about 4.28719. - Robert A. Russell, Oct 21 2014
LINKS
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
EXAMPLE
For n=4 the a(4)=3 solutions are the three achiral (there are no chiral) clusters that can be formed from four simplexes in four-space. One has three attached to a fourth, one has four sharing a common triangle, and the last has neither of these properties.
MATHEMATICA
n = 30;
e[d_, t_]:=Sum[Binomial[d k, k]/((d-1)k+1)t^k, {k, 0, n}]
CoefficientList[Series[(10e[4, t^2]e[2, e[4, t^2]t]^3t
+30e[4, t^4]t(1+e[4, t^4]t)
+20e[1, e[4, t^6] t^2]e[2, e[4, t^6]t^3]t)/60
-(6(e[2, e[4, t^2]t]-1)^2+6e[4, t^4]^2t^2)/24
+(4e[4, t^2]^4t^2+8e[1, e[4, t^6]t^2]e[4, t^6]t^2)/24,
{t, 0, n}]/t, t] (* Robert A. Russell, Apr 23 2012 *)
CROSSREFS
Sequence in context: A148576 A148577 A148578 * A148579 A151263 A148580
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Apr 23 2012
STATUS
approved

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Last modified June 20 13:44 EDT 2024. Contains 373527 sequences. (Running on oeis4.)